Question

Find the value of $k$ if $4x+y+k=0$ is a tangent to the ellipse $x^2+3y^2=3$

Answer 1

Given line and ellipse can be written as $y=-4x-k$ and $\frac{x^2}{3}+\frac{y^2}{1}=1$

Now using condition of tangency, $c^2=a^2{m}^2+b^2$

$\Rightarrow (-k{)}^2=3(-4{)}^2+1^2$

$\Rightarrow k^2=49$

$\therefore k=\pm 7$